Assume that the origin \(O\)  is the center of system (REF ), the
biggest punctured neighborhood enclosing which is called the period
annulus. In what follows we denote it by \(\mathcal {P}\)  and the
projection of which onto \(x-\) axis by \((x_l,x_r)\) . Note that the
annulus \(\mathcal {P}\)  is foliated by periodic orbits. Assigning to
each periodic oval in \(\mathcal {P}\)  its minimal period of the motion
along it, then we define period function \(T\)  of the center. Since
the period function is defined on the set of periodic orbits in
\(\mathcal {P}\) , it is natural firstly to parameterize the set.
Usually we parameterize it by the energy level \(h\)  of the periodic
oval and denote the oval by \(\Gamma _h\)  accordingly. It is known that
for a given continuum of periodic orbits, the number, character
(maximum or minimum) and distribution of critical periods do not
depend on the particular parametrization of the set of periodic
orbits applied. If the period function \(T(h)\)  is constant in a
neighborhood of the center, the center is called isochronous. The
isolated zeros \(\hat{h}\)  of the first order derivative of \(T\)  are
called critical periods, accordingly the periodic oval
\(\Gamma _{\hat{h}}\)  is called critical periodic orbit. The study on
period function is closely related to two-point nonlinear boundary-value problem of ODEs [1]}
and some bifurcation problems
such as subharmonic bifurcations of periodic oscillations and the
bifurcation of steady-state solutions of a reaction-diffusion
equation [2]}, [3]}, [4]}.
